A180662 -id:A180662 - cp's OEIS frontend

A130905 Expansion of e.g.f. exp(x^2 / 2) / (1 - x).

1, 1, 3, 9, 39, 195, 1185, 8295, 66465, 598185, 5982795, 65810745, 789739335, 10266611355, 143732694105, 2155990411575, 34495848612225, 586429426407825, 10555729709800275, 200558864486205225, 4011177290378833575

Offset: 0

Karol A. Penson, Jun 08 2007

nonn

a(n) is also the number of oriented simple graphs on n labeled vertices, such that each weakly connected component with 3 or more vertices is a directed cycle. - Austin Shapiro, Apr 17 2009
The Kn2p sums, p>=1, see A180662 for the definition of these sums, of triangle A193229 lead to this sequence. - Johannes W. Meijer, Jul 21 2011
Compare with A000266 with e.g.f. exp( -x^2 / 2) / (1 - x). - Michael Somos, Jul 24 2011
a(n) is the number of permutations of an n-set where each transposition (two cycle) is counted twice. That is, each transposition is an involution and is its own inverse, but if we imagine each transposition can be oriented in one of two ways, then a permutation with oriented transpositions is just a oriented simple graph. Conversely, an oriented simple graph with restrictions on connected components comes from a permutation with oriented transpositions. - Michael Somos, Jul 25 2011

			1 + x + 3*x^2 + 9*x^3 + 39*x^4 + 195*x^5 + 1185*x^6 + 8295*x^7 + ...
a(2) = 3 because there are 3 oriented simple graphs on two labeled vertices. a(3) = 9 because for oriented simple graphs on three labeled vertices there is 1 with no edges, 6 with one edge, 0 with two edges, and 2 with three edges which are directed cycles such that each weakly connected component with 3 or more vertices is a directed cycle.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.

Cf. A000522, A130906.

Cf. A000266, A355268.

A130905 := proc(n) local x: n!*coeftayl(exp(x^2/2)/(1-x), x=0, n) end: seq(A130905(n), n=0..25); # Johannes W. Meijer, Jul 21 2011

CoefficientList[Series[E^(x^2/2)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( x^2 / 2 + x * O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 24 2011 */

E.g.f.: exp(x^2/2) / (1-x) = exp( x^2 / 2 + sum(k>=1, x^k/k ) ).
E.g.f.: 1/E(0) where E(k)=1 - x/(1 - x/(x + (2*k+2)/E(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
D-finite with recurrence: a(n) = n*a(n-1) + (n-1)*a(n-2) - (n-2)*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ n!*exp(1/2) . - Vaclav Kotesovec, Oct 20 2012
E.g.f.: E(0)/(1-x)^2, where E(k)= 1 - x/(1 - x/(x - 2*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013
a(n) = n! * Sum_{k=0..floor(n/2)} 1/(2^k * k!). - Seiichi Manyama, Feb 27 2024

Superfluous leading 1 deleted by Johannes W. Meijer, Jul 21 2011

A010683 Let S(x,y) = number of lattice paths from (0,0) to (x,y) that use the step set { (0,1), (1,0), (2,0), (3,0), ...} and never pass below y = x. Sequence gives S(n-1,n) = number of 'Schröder' trees with n+1 leaves and root of degree 2.

Original entry on oeis.org

1, 2, 7, 28, 121, 550, 2591, 12536, 61921, 310954, 1582791, 8147796, 42344121, 221866446, 1170747519, 6216189936, 33186295681, 178034219986, 959260792775, 5188835909516, 28167068630713, 153395382655222

Offset: 0

Robert Sulanke (sulanke(AT)diamond.idbsu.edu), N. J. A. Sloane

a(n) is the number of compound propositions "on the negative side" that can be made from n simple propositions.
Convolution of A001003 (the little Schröder numbers) with itself. - Emeric Deutsch, Dec 27 2003
Number of dissections of a convex polygon with n+3 sides that have a triangle over a fixed side (the base) of the polygon. - Emeric Deutsch, Dec 27 2003
a(n-1) = number of royal paths from (0,0) to (n,n), A006318, with exactly one diagonal step on the line y=x. - David Callan, Mar 14 2004
Number of short bushes (i.e., ordered trees with no vertices of outdegree 1) with n+2 leaves and having root of degree 2. Example: a(2)=7 because, in addition to the five binary trees with 6 edges (they do have 4 leaves) we have (i) two edges rb, rc hanging from the root r with three edges hanging from vertex b and (ii) two edges rb, rc hanging from the root r with three edges hanging from vertex c. - Emeric Deutsch, Mar 16 2004
The a(n) equal the Fi2 sums, see A180662, of Schröder triangle A033877. - Johannes W. Meijer, Mar 26 2012
Row sums of A144944 and of A186826. - Reinhard Zumkeller, May 11 2013

T. D. Noe, Table of n, a(n) for n=0..200
A. Bacher, Directed and multi-directed animals on the square lattice with next nearest neighbor edges, arXiv preprint arXiv:1301.1365 [math.CO], 2013-2015. See R(t).
D. Birmajer, J. B. Gil, and M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
Kevin Brown, Hipparchus on Compound Statements, 1994-2010.
Anders Claesson, Giulio Cerbai, Dana C. Ernst, and Hannah Golab, Pattern-avoiding Cayley permutations via combinatorial species, arXiv:2407.19583 [math.CO], 2024.
Shishuo Fu, Zhicong Lin, and Yaling Wang, Refined Wilf-equivalences by Comtet statistics, arXiv:2009.04269 [math.CO], 2020.
Laurent Habsieger, Maxim Kazarian and Sergei Lando, On the second number of Plutarch, Am. Math. Monthly, Vol. 105, No. 5 (May, 1998), p. 446.
H. Kwong, On recurrences of Fahr and Ringel: An Alternate Approach, Fib. Quart., 48 (2010), 363-365; see p. 364.
J. W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
D. G. Rogers and L. W. Shapiro, Deques, trees and lattice paths, in Combinatorial Mathematics VIII: Proceedings of the Eighth Australian Conference. Lecture Notes in Mathematics, Vol. 884 (Springer, Berlin, 1981), pp. 293-303. Math. Rev., 83g, 05038; Zentralblatt, 469(1982), 05005. See Figs. 7a and 8b.
R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
Index entries for sequences related to trees

Cf. A001003, A006318, A033877, A144944, A180662, A186826.
Second right-hand column of triangle A011117.
A177010 has a closely-related g.f..

a010683 = sum . a144944_row  -- Reinhard Zumkeller, May 11 2013

[n le 2 select n else (6*(2*(n-1)^2-1)*Self(n-1) - (n-3)*(2*n-1)*Self(n-2))/((n+1)*(2*n-3)): n in [1..30]]; // G. C. Greubel, Mar 11 2023

a := proc(n) local k: if n=0 then 1 else (2/n)*add(binomial(n, k)* binomial(n+k+1, k-1), k=1..n) fi: end:
seq(a(n), n=0..21); # Johannes W. Meijer, Mar 26 2012, revised Mar 31 2015

f[ x_, y_ ]:= f[ x, y ]= Module[ {return}, If[x==0, return =1, If[y==x-1, return =0, return= f[x,y-1] + Sum[f[k, y], {k,0,x-1} ]]]; return];
(* Do[Print[Table[f[ k, j ], {k, 0, j}]], {j, 10, 0, -1}] *)
Table[f[x, x+1], {x,0,21}]
(* Second program: *)
a[n_] := 2*Hypergeometric2F1[1-n, n+3, 2, -1]; a[0]=1;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 09 2014, after Wolfdieter Lang *)

x='x+O('x^100); Vec(((1-x)^2-(1+x)*sqrt(1-6*x+x^2))/(8*x^2)) \\ Altug Alkan, Dec 19 2015

a = lambda n: (n+1)*hypergeometric([1-n, -n], [3], 2)
[simplify(a(n)) for n in range(22)] # Peter Luschny, Nov 19 2014

G.f.: ((1-t)^2-(1+t)*sqrt(1-6*t+t^2))/(8*t^2) = A(t)^2, with o.g.f. A(t) of A001003.
From Wolfdieter Lang, Sep 12 2005: (Start)
a(n) = (2/n)*Sum_{k=1..n} binomial(n, k)*binomial(n+k+1, k-1).
a(n) = 2*hypergeometric2F1([1-n, n+3], [2], -1), n>=1. a(0)=1. (End)
a(n) = ((2*n+1)*LegendreP(n+1,3) - (2*n+3)*LegendreP(n,3)) / (4*n*(n+2)) for n>0. - Mark van Hoeij, Jul 02 2010
From Gary W. Adamson, Jul 08 2011: (Start)
Let M = the production matrix:
  1, 2, 0, 0, 0, 0, ...
  1, 2, 1, 0, 0, 0, ...
  1, 2, 1, 2, 0, 0, ...
  1, 2, 1, 2, 1, 0, ...
  1, 2, 1, 2, 1, 2, ...
  ...
a(n) is the upper entry in the vector (M(T))^n * [1,0,0,0,...]; where T is the transpose operation. (End)
D-finite with recurrence: (n+2)*(2*n-1)*a(n) = 6*(2*n^2-1)*a(n-1) - (n-2)*(2*n+1)*a(n-2). - Vaclav Kotesovec, Oct 07 2012
a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012
Recurrence (an alternative): (n+2)*a(n) = (4-n)*a(n-4) + 2*(2*n-5)*a(n-3) + 10*(n-1)*a(n-2) + 2*(2*n+1)*a(n-1), n >= 4. - Fung Lam, Feb 18 2014
a(n) = (n+1)*hypergeometric2F1([1-n, -n], [3], 2). - Peter Luschny, Nov 19 2014
a(n) = (A001003(n) + A001003(n+1))/2 = sum(A001003(k) * A001003(n-k), k=0..n). - Johannes W. Meijer, Apr 29 2015

Minor edits by Johannes W. Meijer, Mar 26 2012

A050935 a(n) = a(n-1) - a(n-3) with a(1)=0, a(2)=0, a(3)=1.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, -1, -2, -2, -1, 1, 3, 4, 3, 0, -4, -7, -7, -3, 4, 11, 14, 10, -1, -15, -25, -24, -9, 16, 40, 49, 33, -7, -56, -89, -82, -26, 63, 145, 171, 108, -37, -208, -316, -279, -71, 245, 524, 595, 350, -174, -769, -1119, -945, -176, 943, 1888, 2064, 1121, -767, -2831, -3952

Offset: 1

Richard J. Palmaccio (palmacr(AT)pinecrest.edu), Dec 31 1999

The Ze3 sums, see A180662, of triangle A108299 equal the terms of this sequence without the two leading zeros. - Johannes W. Meijer, Aug 14 2011

R. Palmaccio, "Average Temperatures Modeled with Complex Numbers", Mathematics and Informatics Quarterly, pp. 9-17 of Vol. 3, No. 1, March 1993.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
José L. Ramírez, Víctor F. Sirvent, A note on the k-Narayana sequence, Annales Mathematicae et Informaticae, 45 (2015) pp. 91-105.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1).

When run backwards this gives a signed version of A000931.
Cf. A099529.
Apart from signs, essentially the same as A078013.
Cf. A203400 (partial sums).

a050935 n = a050935_list !! (n-1)
a050935_list = 0 : 0 : 1 : zipWith (-) (drop 2 a050935_list) a050935_list
-- Reinhard Zumkeller, Jan 01 2012

A050935 := proc(n) option remember; if n <= 1 then 0 elif n = 2 then 1 else A050935(n-1)-A050935(n-3); fi; end: seq(A050935(n), n=0..61);

LinearRecurrence[{1,0,-1},{0,0,1},70] (* Harvey P. Dale, Jan 30 2014 *)

a(n)=([0,1,0; 0,0,1; -1,0,1]^(n-1)*[0;0;1])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

From Paul Barry, Oct 20 2004: (Start)
G.f.: x^2/(1-x+x^3).
a(n+2) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k. (End)
G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(12*k-1 + x^2)/( x*(12*k+5 + x^2 ) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 12 2013

Offset adjusted by Reinhard Zumkeller, Jan 01 2012

A154955 a(1) = 1, a(2) = -1, followed by 0, 0, 0, ... .

Original entry on oeis.org

1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Jan 18 2009

Matrix inverse of A000012.
Moebius transform of the sequence A000035. Dirichlet inverse of A209229. Partial sums of a(n) is characteristic function of 1 (A063524). a(n)=(-1)^(n+1)*A019590(n). a(n) for n >= 1 is Dirichlet convolution of following functions b(n), c(n), a(n) = Sum_{d|n} b(d)*c(n/d): a(n) = A000012(n) * A092673(n). Examples of Dirichlet convolutions with function a(n), i.e. b(n) = Sum_{d|n} a(d)*c(n/d): a(n) * A000012(n) = A000035(n), a(n) * A000027(n) = A026741(n), a(n) * A008683(n) = A092673(n), a(n) * A036987(n-1) = A063524(n), a(n) * A000005(n) = A001227(n). - Jaroslav Krizek, Mar 21 2009
The Kn21 sums, see A180662, of triangle A108299 equal the terms of this sequence. - Johannes W. Meijer, Aug 14 2011
{a(n-1)}A132393.%20-%20_Wolfdieter%20Lang">{n>=1}, gives the alternating row sums of A132393. - _Wolfdieter Lang, May 09 2017
With offset 0 the alternating row sums of A097805. - Peter Luschny, Sep 07 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A019590 (absolute values).
Cf. A000005, A000012, A000027, A000035, A001227, A008683, A019590, A026741, A036987, A063524, A092673.
Cf. A097805, A108299, A132393, A180662.

Join[{1, -1}, 0 Range[3, 100]] (* Wesley Ivan Hurt, Jun 22 2024 *)
PadRight[{1,-1},120,0] (* Harvey P. Dale, Dec 10 2024 *)

A154955(n)=(n==1)-(n==2) \\ M. F. Hasler, Jan 13 2012

G.f.: A(x) = x - x^2 = x / (1 + x / (1 - x)). - Michael Somos, Jan 03 2013
a(n) = (2/sqrt(3))*sin((2*Pi/3)*n!). - Lorenzo Pinlac, Jan 16 2022
a(n) = [n = 1] - [n = 2], where [] is the Iverson bracket. - Wesley Ivan Hurt, Jun 22 2024
Multiplicative with a(2) = -1, a(2^e) = 0 if e > 1, a(p^e) = 0 if p > 2. - Antti Karttunen, Dec 17 2024

Keyword:mult added by Antti Karttunen, Dec 17 2024

A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 8, 1, 1, 2, 3, 5, 8, 13, 1, 1, 2, 3, 5, 8, 13, 21, 1, 1, 2, 3, 5, 8, 13, 21, 34, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

Offset: 1

Gary W. Adamson, Mar 23 2005

Triangle of A104762, Fibonacci sequence in each row starts from the right.
The triangle or chess sums, see A180662 for their definitions, link the Fibonacci(n) triangle to sixteen different sequences, see the crossrefs. The knight sums Kn14 - Kn18 have been added. As could be expected all sums are related to the Fibonacci numbers. - Johannes W. Meijer, Sep 22 2010
Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A104763 is reluctant sequence of Fibonacci numbers (A000045), except 0. - Boris Putievskiy, Dec 13 2012

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3;
  1, 1, 2, 3, 5;
  1, 1, 2, 3, 5, 8;
  1, 1, 2, 3, 5, 8, 13; ...

Reinhard Zumkeller, Rows n = 1..100 of table, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A104762, A000045.
Cf. A000071 (row sums). - R. J. Mathar, Jul 22 2009
Triangle sums (see the comments): A000071 (Row1; Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A008346 (Row2); A131524 (Kn11); A001911 (Kn12); A006327 (Kn13); A167616 (Kn14); A180671 (Kn15); A180672 (Kn16); A180673 (Kn17); A180674 (Kn18); A052952 (Kn21 & Kn22 & Kn23 & Fi2 & Ze2); A001906 (Kn3 &Fi1 & Ze3); A004695 (Ca2 & Ze4); A001076 (Ca3 & Ze1); A080239 (Gi2); A081016 (Gi3). - Johannes W. Meijer, Sep 22 2010

Flat(List([1..15], n -> List([1..n], k -> Fibonacci(k)))); # G. C. Greubel, Jul 13 2019

a104763 n k = a104763_tabl !! (n-1) !! (k-1)
a104763_row n = a104763_tabl !! (n-1)
a104763_tabl = map (flip take $ tail a000045_list) [1..]
-- Reinhard Zumkeller, Aug 15 2013

[Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019

Table[Fibonacci[k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 13 2019 *)

for(n=1,15, for(k=1,n, print1(fibonacci(k), ", "))) \\ G. C. Greubel, Jul 13 2019

[[fibonacci(k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019

F(1) through F(n) starting from the left in n-th row.
T(n,k) = A000045(k), 1<=k<=n. - R. J. Mathar, May 02 2008
a(n) = A000045(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
G.f.: (x*y)/((x-1)*(x^2*y^2+x*y-1)). - Vladimir Kruchinin, Jun 21 2025

Edited by R. J. Mathar, May 02 2008

Extended by R. J. Mathar, Aug 27 2008

A129696 Antidiagonal sums of triangular array T defined in A014430: T(j,k) = binomial(j+1, k) - 1 for 1 <= k <= j.

Original entry on oeis.org

1, 2, 5, 9, 17, 29, 50, 83, 138, 226, 370, 602, 979, 1588, 2575, 4171, 6755, 10935, 17700, 28645, 46356, 75012, 121380, 196404, 317797, 514214, 832025, 1346253, 2178293, 3524561, 5702870, 9227447, 14930334, 24157798, 39088150, 63245966

Offset: 1

Paul Curtz, Jun 01 2007

If T is construed as a lower triangular matrix M over the rational field, the inverse M^-1 is a lower triangular matrix containing fractions. Its row sums are the Bernoulli numbers. First column of M^-1 is 1, -1, 2/3, -1/4, -1/30, 1/12, 1/42, -1/12, ... . Multiplied by j! this gives 1, -2, 4, -6, -4, 60, 120, -3660, ... .
The Kn22 sums, see A180662 for the definition of these sums, of the 'Races with Ties' triangle A035317 lead to this sequence. - Johannes W. Meijer, Jul 20 2011
This sequence is the convolution of (1,1,2,3,5,8,13,21,...) and (1,1,2,2,3,3,4,4,5,5,...), i.e., the Fibonacci numbers (A000045) and A008619. - Clark Kimberling, May 28 2012
a(n) is the sum of the first summands over all Arndt compositions of n (see the Checa link). - Daniel Checa, Jan 01 2024

Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Daniel F. Checa, Arndt Compositions: Connections with Fibonacci Numbers, Statistics, and Generalizations, 2023, p. 31.
Larry Ericksen, Problem 7, Problem Section, The Fibonacci Quarterly, Vol. 57, No. 5 (2019), pp. 172, 175-183.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Cf. A000045, A000071, A001924, A008619, A014430, A027641, A035317, A052952 (first differences), A180662.

Cf. A000120, A002487.

m:=36; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=1 to j do M[j, k]:=Binomial(j+1, k)-1; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jun 11 2007

[Fibonacci(n+3)-2-Floor(n/2): n in [1..40]]; // Vincenzo Librandi, Nov 23 2014

with(combinat): a := proc (n) options operator, arrow: fibonacci(n+3)-2-floor((1/2)*n) end proc: seq(a(n), n = 1 .. 34); # Emeric Deutsch, Nov 22 2014

a[n_]:= a[n]= If[n<3, n, a[n-1] + a[n-2] + (n + Mod[n, 2])/2];
Table[a[n], {n,40}] (* Jean-François Alcover, Mar 04 2013 *)
Table[Fibonacci[n+3] -2 -Floor[n/2], {n, 100}] (* Vincenzo Librandi, Nov 23 2014 *)

prpr = 1
prev = 2
for n in range(2,100):
    print(prpr, end=", ")
    curr = prpr+prev + 1 + n//2
    prpr = prev
    prev = curr
# Alex Ratushnyak, Jul 30 2012

[fibonacci(n+3) -2 -(n//2) for n in range(1,41)] # G. C. Greubel, Mar 17 2023

From Paul Barry, Jan 18 2009: (Start)
a(n) = Sum_{k=0..floor(n/2)} A000071(n-2*k+3).
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-2*k} Fibonacci(j+1)). (End)
a(n+1) = a(n-1) + a(n) + 1 + floor(n/2) for n>1, a(0)=1, a(1)=2. - Alex Ratushnyak, Jul 30 2012
From R. J. Mathar, Jul 25 2013: (Start)
G.f.: x/((1 + x)*(1 - x)^2*(1 - x - x^2)).
a(n) + a(n+1) = A001924(n+1). (End)
a(n) = Fibonacci(n+3) - 2 - floor(n/2). - Emeric Deutsch, Nov 22 2014
a(n) = (-5/4 - (-1)^n/4 + (2^(-n)*((1 - t)^n*(-2 + t) + (1 + t)^n*(2 + t)))/t + (-1 - n)/2), where t=sqrt(5). - Colin Barker, Feb 09 2017
E.g.f.: (4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2)) - 5*(4 + x)*cosh(x) - 5*(3 + x)*sinh(x))/10. - Stefano Spezia, Apr 06 2024
a(n) = max_{k = 2^(n+1)..2^(n+2)-1} (A002487(k) - A000120(k)) (Ericksen, 2019). - Amiram Eldar, Jan 30 2025

Edited and extended by Klaus Brockhaus, Jun 11 2007

A120562 Sum of binomial coefficients binomial(i+j, i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 3, 2, 4, 3, 5, 1, 4, 3, 4, 2, 5, 3, 5, 2, 5, 4, 6, 3, 7, 5, 8, 1, 6, 4, 5, 3, 7, 4, 7, 2, 6, 5, 7, 3, 8, 5, 8, 2, 7, 5, 7, 4, 9, 6, 10, 3, 9, 7, 10, 5, 12, 8, 13, 1

Offset: 0

Sam Northshield (samuel.northshield(AT)plattsburgh.edu), Aug 07 2006

a(n) is the number of 'vectors' (..., e_k, e_{k-1}, ..., e_0) with e_k in {0,1,3} such that Sum_{k} e_k 2^k = n. a(2^n-1) = F(n+1)*a(2^{k+1}+j) + a(j) = a(2^k+j) + a(2^{k-1}+j) if 2^k > 4j. This sequence corresponds to the pair (3,1) as Stern's diatomic sequence [A002487] corresponds to (2,1) and Gould's sequence [A001316] corresponds to (1,1). There are many interesting similarities to A000119, the number of representations of n as a sum of distinct Fibonacci numbers.
A120562 can be generated from triangle A177444. Partial sums of A120562 = A177445. - Gary W. Adamson, May 08 2010
The Ca1 and Ca2 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence. Some A120562(2^n-p) sequences, 0 <= p <= 32, lead to known sequences, see the crossrefs. - Johannes W. Meijer, Jun 05 2011

			a(2^n)=1 since a(2n)=a(n).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 6.
Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.
S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.

Cf. A001316 (1,1), A002487 (2,1), A120562 (3,1), A112970 (4,1), A191373 (5,1).
Cf. A177444, A177445. - Gary W. Adamson, May 08 2010
Cf. A000012 (p=0), A000045 (p=1, p=2, p=4, p=8, p=16, p=32), A000071 (p=3, p=6, p=12, p=13, p=24, p=26), A001610 (p=5, p=10, p=20), A001595 (p=7, p=14, p=28), A014739 (p=11, p=22, p=29), A111314 (p=15, p=30), A027961 (p=19), A154691 (p=21), A001911 (p=23). - Johannes W. Meijer, Jun 05 2011
Same recurrence for odd n as A000930.

p := product((1+x^(2^i)+x^(3*2^i)), i=0..25): s := series(p, x, 1000): for k from 0 to 250 do printf(`%d, `, coeff(s, x, k)) od:
A120562:=proc(n) option remember; if n <0 then A120562(n):=0 fi: if (n=0 or n=1) then 1 elif n mod 2 = 0 then A120562(n/2) else A120562((n-1)/2) + A120562((n-3)/2); fi; end: seq(A120562(n),n=0..64); # Johannes W. Meijer, Jun 05 2011

a[0] = a[1] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_?OddQ] := a[n] = a[(n-1)/2] + a[(n-1)/2 - 1]; Table[a[n], {n, 0, 64}] (* Jean-François Alcover, Sep 29 2011 *)
Nest[Append[#1, If[EvenQ@ #2, #1[[#2/2 + 1]], Total@ #1[[#2 ;; #2 + 1]] & @@ {#1, (#2 - 1)/2}]] & @@ {#, Length@ #} &, {1, 1}, 10^4 - 1] (* Michael De Vlieger, Feb 19 2019 *)

Recurrence; a(0)=a(1)=1, a(2*n)=a(n) and a(2*n+1)=a(n)+a(n-1).
G.f.: A(x) = Product_{i>=0} (1 + x^(2^i) + x^(3*2^i)) = (1 + x + x^3)*A(x^2).
a(n-1) << n^x with x = log_2(phi) = 0.69424... - Charles R Greathouse IV, Dec 27 2011

Reference edited and link added by Jason G. Wurtzel, Aug 22 2010

A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 8, 17, 12, 4, 1, 16, 46, 44, 20, 5, 1, 32, 120, 150, 90, 30, 6, 1, 64, 304, 482, 370, 160, 42, 7, 1, 128, 752, 1476, 1412, 770, 259, 56, 8, 1, 256, 1824, 4344, 5068, 3402, 1428, 392, 72, 9, 1, 512, 4352, 12368, 17285, 14000, 7168, 2436

Offset: 1

R. J. Mathar, Feb 21 2010

From Johannes W. Meijer, May 06 2011: (Start)
The Row1, Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Kn3, Kn4 and Ca1 triangle sums link A175136 with several sequences, see the crossrefs. For the definitions of these triangle sums see A180662.
It is remarkable that the coefficients of the right hand columns of A175136, and subsequently those of triangle A175136, can be generated with the aid of the row coefficients of A091894. For the fourth, fifth and sixth right hand columns see A162148, A190048 and A190049. The a(n) formulas of the right hand columns lead to an explicit formula for the T(n,k), see the formulas and the second Maple program. (End)
Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2011.
T(n,k) is the number of noncrossing partitions of n containing k runs, where a block forms a run if it consists of an interval of integers. For example, T(4,2)=6 counts 1/234, 12/34, 123/4, 1/24/3, 13/2/4, 14/2/3. - David Callan, Oct 14 2012

			Triangle starts
    1;
    1,    1;
    2,    2,    1;
    4,    6,    3,    1;
    8,   17,   12,    4,    1;
   16,   46,   44,   20,    5,    1;
   32,  120,  150,   90,   30,    6,    1;
   64,  304,  482,  370,  160,   42,    7,    1;
  128,  752, 1476, 1412,  770,  259,   56,    8,    1;
Triangle (0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  4,  6,  3,  1;
  0,  8, 17, 12,  4,  1; ... - _Philippe Deléham_, Oct 29 2011

David Callan, A bijection on Dyck paths and its cycle structure, El. J. Combinat. 14 (2007) # R28.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
David Callan and Emeric Deutsch, The Run Transform, arXiv:1112.3639 [math.CO], 2011.
K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Nonleft peaks in Dyck paths: a combinatorial approach, Discrete Math., 337 (2014), 97-105.

Triangle columns: A000012, A000027, A002378, A162148, A190048, A190049, A011782, A190050, A190051.
Triangle sums (see the comments): A000108 (Row1), A005043 (Related to Kn11, Kn12, Kn13 and Kn4), A007477 (Related to Kn21, Kn22, Kn23 and Kn3), A099251 (Kn4), A166300 (Ca1). - Johannes W. Meijer, May 06 2011
Cf. A000108 (row sums), A196182

lco := proc(siz,leav) (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/2/x ; coeftayl(%,x=0,siz ) ; coeftayl(%,y=0,leav ) ; end proc: seq(seq(lco(n,k),k=1..n),n=1..9) ;
T := proc(n, k): add(A091894(n-k, k1)*binomial(n-k1-1, n-k), k1=0..floor((n-k)/2)) end: A091894 := proc(n, k): if n=0 and k=0 then 1 elif n=0 then 0 else 2^(n-2*k-1)* binomial(n-1, 2*k) * binomial(2*k, k)/(k+1) fi end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, May 06 2011, revised Nov 23 2012

A091894[n_, k_] := 2^(n - 2*k - 1)*Binomial[n - 1, 2*k]*(Binomial[2*k, k]/(k + 1)); t[n_, k_] := Sum[A091894[n - k, k1]*Binomial [n - k1 - 1, n - k], {k1, 0, (n - k)/2}]; t[n_, n_] = 1; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten(* Jean-François Alcover, Jun 13 2013, after Johannes W. Meijer *)

G.f.: (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/(2*x).

T(n,k) = Sum_{k1=0..floor((n-k)/2)} A091894(n-k, k1)*binomial(n-k1-1, n-k), 1 <= k <= n. - Johannes W. Meijer, May 06 2011

Variable names changed by Johannes W. Meijer, May 06 2011

A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 127, 322, 826, 2135, 5545, 14445, 37701, 98514, 257608, 673933, 1763581, 4615823, 12082291, 31628466, 82798926, 216761547, 567474769, 1485645049, 3889431721, 10182603746, 26658304492, 69792188337, 182718064101, 478361686155, 1252366480135

Offset: 0

N. J. A. Sloane

Number of block fountains with exactly n coins in the base when mirror image fountains are identified. - Michael Woltermann (mwoltermann(AT)washjeff.edu), Oct 06 2010
a(n) = C(F(n+1)+1,2) + C(F(n)+1,2) = pairwise sums of A033192. - Ralf Stephan, Jul 06 2003
Number of (3412,54312)- and (3412,45321)-avoiding involutions in S_{n+1}. - Ralf Stephan, Jul 06 2003
Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. - Herbert Kociemba, May 31 2004
The sequence 1,1,2,4,9,... has g.f. 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x))))=(1-3*x+x^2+x^2)/(1-4*x+3*x^2+2*x^3-x^4), and general term (A001519(n)+A000045(n+1))/2. It is the binomial transform of A001519 aerated. - Paul Barry, Dec 17 2009
The Kn3 and Kn4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence. - Johannes W. Meijer, Jul 14 2011
Convolution of [1,1,1,2,5,...], which is A001519 with another leading 1, and A212804. - R. J. Mathar, Apr 14 2018
a(n) is the number of Motzkin n-paths of height <= 3. - Alois P. Heinz, Nov 24 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..300 from Vincenzo Librandi)
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8, arXiv:math/0307050 [math.CO], 2003.
S. Felsner, D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392. (Annotated scanned copy)
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv preprint arXiv:1105.3713 [math.CO], 2011.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Michael Woltermann, Problem 1826, Mathematics Magazine, 83 (2010), 304-305.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000045, A001006, A001519, A131719.

A005207:=-(1-2*z-z^2+z^3)/(z^2-3*z+1)/(z^2+z-1); # Simon Plouffe in his 1992 dissertation with offset 0
a:= n-> (Matrix([[1,1,1,3]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [4,-3,-2,1][i] else 0 fi)^n)[1,2]: seq(a(n), n=0..34); # Alois P. Heinz, Sep 06 2008

LinearRecurrence[{4, -3, -2, 1}, {1, 2, 4, 9}, 30] (* Jean-François Alcover, Jan 31 2016 *)

a(n)=(fibonacci(2*n-1)+fibonacci(n+1))/2

x='x+O('x^50); Vec(-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1))) \\ G. C. Greubel, Mar 05 2017

G.f.: 1-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1)).
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).
a(n) = (w^(2*n-1) + w^(1-2*n) + w^(n+1) - (-w)^(-1-n))/(4*w-2) where w = (1+sqrt(5))/2.
a(n) = (2/5)*Sum_{k=1..4} ( sin(Pi*k/5)^2*(1 + 2*cos(Pi*k/5))^n ). - Herbert Kociemba, May 31 2004
a(-1-2*n) = A027994(2*n); a(-2*n)=A059512(2*n+1).
Let M = an infinite tridiagonal matrix with all 1's in the super and main diagonals and [1,1,1,0,0,0,...] in the subdiagonal. Let V = vector [1,0,0,0,...]. The sequence is generated as leftmost column of M*V iterates. - Gary W. Adamson, Jun 07 2011
2*a(n) = A000045(n+1) + A001519(n). - R. J. Mathar, Apr 14 2018
a(n) mod 2 = A131719(n+3). - Alois P. Heinz, Nov 24 2023

a(0)=1 prepended by Alois P. Heinz, Nov 24 2023

A096338 a(n) = (2/(n-1))a(n-1) + ((n+5)/(n-1))a(n-2) with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 6, 10, 20, 30, 50, 70, 105, 140, 196, 252, 336, 420, 540, 660, 825, 990, 1210, 1430, 1716, 2002, 2366, 2730, 3185, 3640, 4200, 4760, 5440, 6120, 6936, 7752, 8721, 9690, 10830, 11970, 13300, 14630, 16170, 17710, 19481, 21252, 23276, 25300, 27600

Offset: 0

Benoit Cloitre, Jun 28 2004

Without the leading zero, Poincaré series [or Poincare series] P(C_{2,2}; t).
Starting (1, 2, 6, ...) = partial sums of the tetrahedral numbers, A000292 with repeats: (1, 1, 4, 4, 10, 10, 20, 20, 35, 35, ...). - Gary W. Adamson, Mar 30 2009
Starting with 1 = [1, 2, 3, ...] convolved with the aerated triangular series, [1, 0, 3, 0, 6, ...]. - Gary W. Adamson, Jun 11 2009
From Alford Arnold, Oct 14 2009: (Start)
a(n) is also related to Dyck Paths. Note that
0 1 2 6 10 20 30 50 70 105 ...
minus
0 0 0 0 1 2 6 10 20 30 ...
equals
0 1 2 6 9 18 24 40 50 75 ... A028724
(End)
The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums of A139600 are related to the sequence given above; e.g., Ze1(n) = 3*A096338(n-1) - 3*A096338(n-3) + 9*A096338(n-4), with A096338(n) = 0 for n <= -1. For the definition of these triangle sums, see A180662. - Johannes W. Meijer, Apr 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dragomir Z. Djokovic, Poincaré series [or Poincare series] of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 3.
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4, 19.
M. Navascues and T. Vertesi, The Structure of Matrix Product States, arXiv preprint arXiv:1509.04507 [quant-ph], 2015-2018.
Miguel Navascués and Tamás Vértesi, Bond dimension witnesses and the structure of homogeneous matrix product states, Quantum 2 (2018): 50.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A006918, A038163, A060099, A058187, A000292, A028724.

A096338:=n->-(floor(n/2)+1)*(floor(n/2)+2)*(floor(n/2)+3)*(3*floor(n/2)-2*n)/12; seq(A096338(k),k=0..100); # Wesley Ivan Hurt, Oct 04 2013

t = {0, 1}; Do[AppendTo[t, (2/(n - 1))*t[[-1]] + ((n + 5)/(n - 1))*t[[-2]]], {n, 2, 50}]; t (* T. D. Noe, Oct 08 2013 *)
CoefficientList[Series[x/((1 - x)^2*(1 - x^2)^3), {x, 0, 45}], x] (* or *)
Nest[Append[#1, (2/(#2 - 1))*#1[[-1]] + ((#2 + 5)/(#2 - 1))*#1[[-2]] ] & @@ {#, Length@ #} &, {0, 1}, 44] (* Michael De Vlieger, May 30 2018 *)

G.f.: x/((1-x)^2*(1-x^2)^3). - Ralf Stephan, Jun 29 2004
a(n) = Sum_{k=1..floor(n/2)+1} ( Sum_{i=1..k} i*(n-2*k+2) ) = -(floor(n/2)+1) * (floor(n/2)+2) * (floor(n/2)+3) * (3*floor(n/2) - 2*n)/12. - Wesley Ivan Hurt, Sep 26 2013
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8). - Wesley Ivan Hurt, Nov 26 2020
128*a(n) = 8*n^3 +94/3*n^2 +44*n +15 +2/3*n^4 -2*(-1)^n*n^2 -12*(-1)^n*n -15*(-1)^n. - R. J. Mathar, Mar 23 2021

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A130905 Expansion of e.g.f. exp(x^2 / 2) / (1 - x).

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A010683 Let S(x,y) = number of lattice paths from (0,0) to (x,y) that use the step set { (0,1), (1,0), (2,0), (3,0), ...} and never pass below y = x. Sequence gives S(n-1,n) = number of 'Schröder' trees with n+1 leaves and root of degree 2.

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A050935 a(n) = a(n-1) - a(n-3) with a(1)=0, a(2)=0, a(3)=1.

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A154955 a(1) = 1, a(2) = -1, followed by 0, 0, 0, ... .

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A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.

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A129696 Antidiagonal sums of triangular array T defined in A014430: T(j,k) = binomial(j+1, k) - 1 for 1 <= k <= j.

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A096338 a(n) = (2/(n-1))a(n-1) + ((n+5)/(n-1))a(n-2) with a(0)=0 and a(1)=1.